El e c t ro n ic
Jo ur
n a l o
f P
r o
b a b il i t y
Vol. 15 (2010), Paper no. 63, pages 1938–. Journal URL
http://www.math.washington.edu/~ejpecp/
The center of mass for spatial branching processes and an
application for self-interaction
János Engländer
∗Abstract
Consider the center of mass of ad-dimensional supercritical branching-Brownian motion. We first show that it is a Brownian motion being slowed down such that it tends to a limiting position almost surely, and that this is also true for a model where branching Brownian motion is modified byattraction/repulsion between particles, where the strength of the attraction/repulsion is given by the parameterγ6=0.
We then put this observation together with the description of the interacting system as viewed from its center of mass, and get our main result in Theorem 16: Ifγ >0 (attraction), then, as
n→ ∞,
2−nZn(dy) w
⇒
γ
π
d/2
exp−γ|y−x|2dy, Px−a.s.
for almost allx∈Rd, whereZ(dy)denotes the discrete measure-valued process corresponding
to the interacting branching particle system,⇒w denotes weak convergence, andPxdenotes the law of the particle system conditioned to havexas the limit for the center of mass.
A conjecture is stated regarding the behavior of the local mass in the repulsive case.
We also consider a supercritical super-Brownian motion, and show that, conditioned on survival, its center of mass is a continuous process having an a.s. limit ast→ ∞.
Key words: Branching Brownian motion, super-Brownian motion, center of mass, self-interaction, Curie-Weiss model, McKean-Vlasov limit, branching Ornstein-Uhlenbeck process, spatial branching processes,H-transform.
∗Department of Mathematics, University of Colorado at Boulder, Boulder, CO-80309, USA.
AMS 2000 Subject Classification:Primary 60J60; Secondary: 60J80.
1
Introduction
1.1
Notation
1. Probability:The symbolE∁will denote the complement of the eventE, andX⊕Y will denote the independent sum of the random variablesX andY.
2. Topology and measures:The boundary of the setBwill be denoted by∂Band the closure of Bwill be denoted by cl(B), that is cl(B):=B∪∂B; the interior ofBwill be denoted by ˙BandBε will denote theε-neighborhood ofB. We will also use the notation ˙Bε:={y ∈B: Bε+y⊂B}, whereB+b:={y : y−b∈B}andBt :={x∈Rd : |x|<t}. By abounded rational rectangle
we will mean a setB ⊂Rd of the formB =I
1×I2× · · · ×Id, where Ii is a bounded interval
with rational endpoints for each 1≤i≤d. The family of all bounded rational rectangles will be denoted byR.
Mf(Rd) andM1(Rd) will denote the space of finite measures and the space of probability
measures, respectively, onRd. For µ ∈ M
f(Rd), we definekµk := µ(Rd). |B| will denote
the Lebesgue measure ofB. The symbols “⇒w” and“⇒v ” will denote convergence in the weak topology and in the vague topology, respectively.
3. Functions:For f,g>0, the notation f(x) =O(g(x))will mean that f(x)≤C g(x)if x> x0 with somex0∈R,C >0; f ≈gwill mean that f/gtends to 1 given that the argument tends to an appropriate limit. For N → R functions the notation f(n) = Θ(g(n)) will mean that c≤ f(n)/g(n)≤C ∀n, with some c,C >0.
4. Matrices:The symbolId will denote the d-dimensional unit matrix, and r(A)will denote the rank of a matrixA.
5. Labeling:In this paper we will often talk about the ‘ithparticle’ of a branching particle system. By this we will mean that we label the particles randomly, but in a way that does not depend on their spatial position.
1.2
A model with self-interaction
Consider a dyadic (i.e. precisely two offspring replaces the parent) branching Brownian motion (BBM) inRd with unit time branching and with the following interaction between particles: if Z
denotes the process andZti is theithparticle, thenZti ‘feels’ the drift
1 nt
X
1≤j≤nt
γ·Ztj− ·,
whereγ6=0 , that is the particle’s infinitesimal generator is
1 2∆ +
1
nt
X
1≤j≤nt
γ·
Ztj−x
· ∇. (1.1)
(Here and in the sequel,nt is a shorthand for 2⌊t⌋, where⌊t⌋is the integer part oft.) Ifγ >0, then
To be a bit more precise, we can define the process by induction as follows. Z0 is a single particle at the origin. In the time interval[m,m+1)we define a system of 2m interacting diffusions, starting at the position of their parents at the end of the previous step (at timem−0) by the following system of SDE’s:
dZti=dWtm,i+ γ 2m
X
1≤j≤2m
(Ztj−Zti)dt; i=1, 2, . . . , 2m, (1.2)
whereWm,i,i=1, 2, . . . , 2m;m=0, 1, ... are independent Brownian motions.
Remark 1 (Attractive interaction). If there were no branching and the interval [m,m+1) were extended to [0,∞), then for γ > 0 the interaction (1.2) would describe the ferromagnetic Curie-Weiss model, a model appearing in the microscopic statistical description of a spatially homogeneous gas in a granular medium. It is known that asm→ ∞, a Law of Large Numbers, theMcKean-Vlasov limitholds and the normalized empirical measure
ρm(t):=2−m
2m X
i=1
δZi t
tends to a probability measure-valued solution of
∂ ∂tρ=
1 2∆ρ+
γ
2∇ · ρ∇f
ρ,
where fρ(x):=RRd|x−y|
2ρ(d y). (See p. 24 in[8]and the references therein.)
⋄
Remark 2(More general interaction). It seems natural to replace the linearity of the interaction by a more general rule. That is, to define and analyze the system where (1.2) is replaced by
dZti=dWtm,i+2−m X 1≤j≤2m
g(|Ztj−Zti|) Z
j t−Zti
|Ztj−Zi t|
dt; i=1, 2, . . . , 2m,
where the function g :R+ →R has some nice properties. (In this paper we treat the g(x) =γx
case.) This is part of a future project with J. Feng. ⋄
1.3
Existence and uniqueness
Notice that the 2minteracting diffusions on[m,m+1)can be considered as a single 2md-dimensional Brownian motion with linear (and therefore Lipschitz) driftb:R2md
→R2md
:
b x1,x2, ...,xd,x1+d,x2+d, ...,x2d, ...,x1+(2m−1)d,x2+(2m−1)d, ...,x2md) =:γ(β1,β2, ...,β2md)T,
where
βk=2−m(xbk+xbk+d+...+xbk+(2m−1)d)−xk, 1≤k≤2md,
1.4
Results on the self-interacting model
We are interested in the long time behavior of Z, and also whether we can say something about the number of particles in a given compact set for nlarge. In the sequel we will use the standard notation〈g,Zt〉=〈Zt,g〉:=Pnt
i=1g(Z
i t).
In this paper we will first show that Z asymptotically becomes a branching Ornstein-Uhlenbeck process (inward for attraction and outward for repulsion), but
1. the origin is shifted to a random point which hasd-dimensional normal distributionN(0, 2Id), and
2. the Ornstein-Uhlenbeck particles are not independent but constitute a system with a degree of freedom which is less than their number by precisely one.
The main result of this articleconcerns the local behavior of the system: we will prove a scaling limit theorem (Theorem 16) for the local mass whenγ >0 (attraction), and formulate and motivate a conjecture (Conjecture 18) whenγ <0 (repulsion).
1.5
The center of mass for supercritical super-Brownian motion
In Lemma 6 we will show that Zt := n1 t
Pnt
i=1Zti, the center of mass for Z satisfies limt→∞Zt =N,
whereN ∼ N(0, 2Id). In fact, the proof will reveal that Zmoves like a Brownian motion, which is nevertheless slowed down tending to a final limiting location (see Lemma 6 and its proof).
Since this is also true forγ=0 (BBM with unit time branching and no self-interaction), our first nat-ural question is whether we can prove a similar result for the supercritical super-Brownian motion.
Let X be the (1
2∆,β,α;R
d)-superdiffusion with α,β > 0 (supercritical super-Brownian motion).
Hereβ is the ‘mass creation parameter’ or ‘mass drift’, whileα > 0 is the ‘variance (or intensity) parameter’ of the branching — see[6]for more elaboration and for a more general setting.
Let Pµ denote the corresponding probability when the initial finite measure isµ. (We will use the abbreviationP:=Pδ0.) Let us restrictΩto the survival set
S:={ω∈Ω|Xt(ω)>0, ∀t>0}.
Sinceβ >0,Pµ(S)>0 for allµ6=0.
It turns out that on the survival set the center of mass forX stabilizes:
Theorem 3. Let α,β > 0 and let X denote the center of mass process for the (12∆,β,α;Rd)
-superdiffusion X , that is let
X:= 〈id,X〉 kXk ,
where〈f,X〉:=RRd f(x)X(dx)andid(x) = x. Then, on S, X is continuous and converges P-almost
Remark 4. A heuristic argument for the convergence is as follows. Obviously, the center of mass is invariant underH-transforms1 wheneverH is spatially (but not temporarily) constant. LetH(t):= e−βt. ThenXH is a(1
2∆, 0,e
−βtα;Rd)-superdiffusion, that is, a critical super-Brownian motion with
a clock that is slowing down. Therefore, heuristically it seems plausible thatXH, the center of mass
for the transformed process stabilizes, because after some large timeT, if the process is still alive, it behaves more or less like the heat flow (e−βtα is small), under which the center of mass does not
move. ⋄
1.6
An interacting superprocess model
The next goal is to construct and investigate the properties of a measure-valued process with repre-sentative particles that are attracted to or repulsed from its center of mass.
There is one work in this direction we are aware of: motivated by the present paper, H. Gill [9] has constructed very recently asuperprocess with attraction to its center of mass. More precisely, Gill constructs a supercritical interacting measure-valued process with representative particles that are attracted to or repulsed from its center of mass using Perkins’shistorical stochastic calculus.
In the attractive case, Gill proves the equivalent of our Theorem 16 (see later): on S, the mass normalized process converges almost surely to the stationary distribution of the Ornstein-Uhlenbeck process centered at the limiting value of its center of mass; in the repulsive case, he obtains sub-stantial results concerning the equivalent of our Conjecture 18 (see later), using[7]. In addition, a version of Tribe’s result[12]is presented in[9].
2
The mass center stabilizes
Returning to the discrete interacting branching system, notice that
1 nt
X
1≤j≤nt
Ztj−Zti
=Zt−Zti, (2.1)
and sothe net attraction pulls the particle towards the center of mass(net repulsion pushes it away from the center of mass).
Remark 5. The reader might be interested in the very recent work[11], where a one dimensional particle system is considered with interaction via the center of mass. There is a kind of attraction towards the center of mass in the following sense: each particle jumps to the right according to some common distributionF, but the rate at which the jump occurs is a monotone decreasing function of the signed distance between the particle and the mass center. Particles being far ahead slow down,
while the laggards catch up. ⋄
Since the interaction is in fact through the center of mass, the following lemma is relevant:
Lemma 6(Mass center stabilizes). There is a random variable N∼ N(0, 2Id)such thatlimt→∞Zt=
N a.s.
Proof. Letm∈N. Fort∈[m,m+1)there are 2m particles moving around and particleZi
t’s motion
is governed by
dZti=dWtm,i+γ(Zt−Zti)dt.
Since 2mZt=P2i=m1Zti, we obtain that
dZt=2−m
2m X
i=1
dZti=2−m 2m X
i=1
dWtm,i+ γ 2m 2
mZ t−
2m X
i=1 Zti
!
dt =2−m 2m X
i=1
dWtm,i.
So, for 0≤τ <1,
Zm+τ=Zm+2−m
2m
M
i=1
Wτm,i =:Zm+2−m/2B(m)(τ), (2.2)
where we note thatB(m)is a Brownian motion on[m,m+1). Using induction, we obtain that2
Zt=B(0)(1)⊕
1 p
2B (1)(1)
⊕ · · · ⊕ 1 2k/2B
(k)(1) ⊕
· · · ⊕p 1
2⌊t⌋−1
B(⌊t⌋−1)(1)⊕p1 ntB
(⌊t⌋)(τ), (2.3)
whereτ:=t− ⌊t⌋.
By Brownian scalingW(m)(·):=2−m/2B(m)(2m·), m≥1 are (independent) Brownian motions. We have
Zt=W(0)(1)⊕W(1)
1
2
⊕ · · · ⊕W(⌊t⌋−1)
1
2⌊t⌋−1
⊕W(⌊t⌋)
τ
nt
.
By the Markov property, in fact
Zt=Wc
1+1
2+· · ·+ 1
2⌊t⌋−1 +
τ
nt
,
whereWcis a Brownian motion (the concatenation of theW(i)’s), and sinceWchas a.s. continuous paths, limt→∞Zt=Wc(2), a.s.
For another proof see the remark after Lemma 9.
Remark 7. It is interesting to note thatZ is in fact a Markov process. To see this, let{Ft}t≥0 be the canonical filtration for Z and{Gt}t≥0 the canonical filtration for Z. SinceGs ⊂ Fs, it is enough to
check the Markov property withGs replaced byFs.
Assume first 0≤s< t,⌊s⌋=⌊t⌋=:m. Then the distribution of Zt conditional onFsis the same as conditional on Zs, because Z itself is a Markov process. But the distribution ofZt only depends on Zs throughZs, as
Zt d
=Zs⊕W(2
m)t−s
2m
, (2.4)
whatever Zs is. That is, P(Zt∈ · | Fs) =P(Zt ∈ · |Zs) = P(Zt ∈ · |Zs). Note that this is even true whens∈Nandt=s+1, becauseZt=Zt−0.
Assume now thats<⌊t⌋=:m. Then the equation P(Zt ∈ · | Fs) = P(Zt ∈ · | Zs), is obtained by
conditioning successively onm,m−1, ...,⌊s⌋+1,s. ⋄
We will also need the following fact later.
Lemma 8. The coordinate processes of Z are independent one dimensional interactive branching pro-cesses of the same type as Z.
We leave the simple proof to the reader.
3
Normality via decomposition
Let againm:=⌊t⌋. We will need the following result.
Lemma 9 (Decomposition). In the time interval [m,m + 1) the d · 2m-dimensional process (Zt1,Zt2, ...,Zt2m) can be decomposed into two components: a d-dimensional Brownian motion and an independent d(2m−1)-dimensional Ornstein-Uhlenbeck process with parameterγ.
More precisely, each coordinate process (as a 2m-dimensional process) can be decomposed into two components: a one-dimensional Brownian motion in the direction(1, 1, ..., 1)and an independent(2m− 1)-dimensional Ornstein-Uhlenbeck process with parameter γ in the ortho-complement of the vector (1, 1, ..., 1).
Furthermore,(Zt1,Zt2, ...,Znt
t )is d nt-dimensional joint normal for all t≥0.
Proof of lemma.By Lemma 8, we may assume thatd=1. We prove the statement by induction.
(i) Form=0 it is trivial.
(ii) Suppose that the statement is true form−1. Consider the timemposition of the 2m−1particles (Zm1,Zm2, ...,Zm2m−1)‘just before’ the fission. At the instant of the fission we obtain the 2m-dimensional vector
(Zm1,Zm1,Zm2,Zm2, ...,Zm2m−1,Zm2m−1),
which has the same distribution on the 2m−1 dimensional subspace S:={x∈R2m
|x1=x2,x3= x4, ...,x2m−1=x2m}
of R2m as the vector p2(Z1
m,Zm2, ...,Z2 m−1
m ) on R2
m−1
. Since, by the induction hypothesis, (Zm1,Zm2, ...,Zm2m−1) is normal, the vector formed by the particle positions ‘right after’ the fission is a 2m-dimensional degenerate normal. (The reader can easily visualize this form=1: the distribu-tion of(Z11,Z11)is clearlyp2 times the distribution of a Brownian particle at time 1, i.e. N(0,p2) on the line x1=x2.)
Since the convolution of normals is normal, therefore, by the Markov property, it is enough to prove the statement when the 2m particles start at the origin and the clock is reset: t∈[0, 1).
Define the 2m-dimensional processZ∗on the time intervalt∈[0, 1)by Zt∗:= (Z1t,Zt2, ...,Z2tm),
starting at the origin. Because the interaction between the particles attracts the particles towards the center of mass,Z∗is a Brownian motion with drift
γ(Zt,Zt, ...,Zt)−(Zt1,Z
2
t, ...,Z
2m
t )
Notice that this drift is orthogonal to the vector3 v:= (1, 1, ..., 1), that is, the vector (Zt,Zt, ...,Zt) is nothing but the orthogonal projection of(Zt1,Zt2, ...,Zt2m)to the line ofv. This observation imme-diately leads to the following decomposition. The process Z∗can be decomposed into two compo-nents:
• the component in the direction ofvis a Brownian motion
• in S, the ortho-complement of v, it is an independent Ornstein-Uhlenbeck process with pa-rameterγ.
Remark 10. (i) The proof also shows that (Zt1,Z2t, ...,Znt
t ) given Zs is a.s. joint normal for all
t>s≥0.
(ii) Consider the Brownian component in the decomposition appearing in the proof. Since, on the other hand, this coordinate is 2m/2Zt, using Brownian scaling, one obtains a slightly different way of
seeing thatZt stabilizes at a position which is distributed as the time 1+2−1+2−2+...+2−m+...=2 value of a Brownian motion. (The decomposition shows this ford =1 and then it is immediately
upgraded to generald by independence.) ⋄
Corollary 11 (Asymptotics for finite subsystem). Let k ≥ 1 and consider the subsystem (Zt1,Zt2, ...,Ztk), t ≥ m0 for m0 := ⌊log2k⌋+1. (This means that at time m0 we pick k particles and at every fission replace the parent particle by randomly picking one of its two descendants.) Let the real numbers c1, ...,cksatisfy
k
X
i=1 ci=0,
k
X
i=1
ci2=1. (3.1)
DefineΨt = Ψ(c1,...,ck)
t :=
Pk
i=1ciZti and note thatΨt is invariant under the translations of the
coordi-nate system. LetLt denote its law.
For every k≥1and c1, ...,ck satisfying (3.1),Ψ(c1,...,ck)is the same d-dimensional Ornstein-Uhlenbeck
process corresponding to the operator1/2∆−γ∇ ·x, and in particular, forγ >0,
lim
t→∞Lt=N
0, 1 2γId
.
For example, takingc1=1/p2,c2=−1/p2, we obtain that when viewed from a tagged particle’s position, any given other particle moves asp2 times the above Ornstein-Uhlenbeck process. Proof. By independence (Lemma 8) it is enough to consider d = 1. For m fixed, consider the decomposition appearing in the proof of Lemma 9 and recall the notation. By (3.1), whatever m≥m0is, the 2m dimensional unit vector
(c1,c2, ...,ck, 0, 0, ..., 0)
is orthogonal to the 2m dimensional vectorv. This means thatΨ(c1,...,ck) is a one dimensional
pro-jection of the Ornstein-Uhlenbeck component ofZ∗, and thus it is itself a one dimensional Ornstein-Uhlenbeck process (with parameterγ) on the unit time interval.
Now, although as m grows, the Ornstein-Uhlenbeck components of Z∗ are defined on larger and larger spaces (S ⊂ R2m
is a 2m−1 dimensional linear subspace), the projection onto the direction of(c1,c2, ...,ck, 0, 0, ..., 0) is always the same one dimensional Ornstein-Uhlenbeck process, i.e. the
different unit time ‘pieces’ ofΨ(c1,...,ck)obtained by those projections may be concatenated.
4
The interacting system as viewed from the center of mass
Recall that by (2.2) the interaction has no effect on the motion of Z. Let us see now how the interacting system looks like when viewed fromZ.
4.1
The description of a single particle
Using our usual notation, assume thatt∈[m,m+1)and letτ:=t− ⌊t⌋. When viewed from Z, the relocation4 of a particle is governed by
d(Z1t −Zt) =dZt1−dZt=dWtm,1−2−m
2m X
i=1
dWtm,i−γ(Z1t −Zt)dt.
So ifY1:=Z1−Z, then
dYt1=dWtm,1−2−m 2m
X
i=1
dWtm,i−γYt1dt.
Clearly,
Wτm,1−2−m 2m M
i=1
Wτm,i = 2m M
i=2
2−mWτm,i⊕(1−2−m)Wτm,1;
and, by a trivial computation, the right hand side is a Brownian motion with mean zero and variance (1−2−m)τId :=σ2mτId. That is,
dYt1=σmdWf m,1
t −γYt1dt,
whereWfm,1is a standard Brownian motion.
We have thus obtained that on the time interval [m,m+1), Y1 corresponds to the Ornstein-Uhlenbeck operator
1 2σ
2
m∆−γx· ∇. (4.1)
Since formlargeσmis close to one, the relocation viewed from the center of mass isasymptotically
governed by an O-U process corresponding to 1
2∆−γx· ∇.
Remark 12(Asymptotically vanishing correlation between driving BM’s). LetfWm,i,k be thekth co-ordinate of theithBrownian motion:Wfm,i= (Wfm,i,k,k=1, 2, ...,d)andBm,i,kbe thekthcoordinate ofWm,i. For 1≤i6= j≤2m, we have
EσmWfτm,i,k·σmWfτm,j,k
=
E
Bτm,i,k−2−m 2m M
r=1 Bτm,r,k
Bτm,j,k−2−m 2m M
r=1 Bmτ,r,k
=
that is, fori6= j,
EfWτm,i,kWfτm,j,ℓ=− δkℓ
2m−1τ. (4.2)
Hencethe pairwise correlation tends to zeroas t→ ∞(recall thatm=⌊t⌋andτ=t−m∈[0, 1)).
And of course, for the variances we have
EWfτm,i,kWfτm,i,ℓ=δkℓ·τ, for 1≤i≤2m. ⋄ (4.3)
4.2
The description of the system; the ‘degree of freedom’
Fixm≥1 and fort ∈[m,m+1)let Yt:= (Yt1, ...,Yt2m)T, where()T denotes transposed. (This is a vector of length 2m where each component itself is ad dimensional vector; one can actually view it as a 2m×dmatrix too.) We then have
dYt=σmdWf
(m)
t −γYtdt,
where
f
W(m)=Wfm,1, ...,Wfm,2mT
and
f
Wτm,i=σ−m1
Wτm,i−2−m 2m M
j=1 Wτm,j
, i=1, 2, ..., 2m
are mean zero Brownian motions with correlation structure given by (4.2)-(4.3).
Just like at the end of Subsection 1.2, we can considerY as a single 2md-dimensional diffusion. Each of its components is an Ornstein-Uhlenbeck process with asymptotically unit diffusion coefficient.
By independence, it is enough to consider thed=1 case, and so from now on, in this subsection we assume thatd=1.
Let us first describe the distribution ofWft(m)fort≥0 fixed. Recall that{Wsm,i, s≥0; i=1, 2, ..., 2m} are independent Brownian motions. By definition,Wft(m)is a 2m-dimensional multivariate normal:
f
Wt(m)=σ−m1·
1−2−m −2−m ... −2−m −2−m 1−2−m ... −2−m
. . .
−2−m −2−m ... 1−2−m
Wt(m)=:σm−1A(m)Wt(m),
whereWt(m)= (Wtm,1, ...,Wtm,2m)T, yielding
dYt=A(m)dWt(m)−γYtdt.
Since we are viewing the system from the center of mass, Wft(m) is asingular multivariate normal and thusY is a degenerate diffusion. The ‘true’ dimension ofWft(m)is r(A(m)).
Proof. We will simply writeAinstead ofA(m). Since the columns ofAadd up to zero, the matrixA is not of full rank: r(A)≤2m−1. On the other hand,
2mA+
1 1 ... 1 1 1 ... 1
. . .
1 1 ... 1
=2mI,
whereIis the 2m-dimensional unit matrix, and so by subadditivity,
r(A) +1=r(2mA) +1≥2m.
By Lemma 13,Wft(m)is concentrated onS, and there the vectorWft(m) hasnon-singularmultivariate normal distribution.5 What this means is that even thoughWfm,1, ...,Wfm,2m are not independent, their ‘degree of freedom’ is 2m−1, i. e. the 2m-dimensional vector Wft(m) isdetermined by2m−1 independent components(corresponding to 2m−1 principal axes).
5
Asymptotic behavior
5.1
Conditioning
How can one put together thatZttends to a random final position with the description of the system ‘as viewed from Zt?’ The following lemma is the first step in this direction.
Lemma 14(Independence). LetT be the tailσ-algebra of Z.
1. For t≥0, the random vector Yt is independent of the path{Zs}s≥t.
2. The process Y = (Yt;t≥0)is independent ofT.
Proof. In both parts we will refer to the following fact. Lets≤t,s∈[mb,mb+1);t∈[m,m+1)with
b
m≤m. Since the random variables Z1t,Zt2, ...,Zt2m are exchangeable, thus, denoting bn:=2mb,n:= 2m, the vectors Zt andZs1−Zs are uncorrelated for 0≤s≤t. Indeed, by Lemma 8 we may assume thatd=1 and then
E[Zt·Zs1−Zs] =E Z
1
t +Z
2
t +...+Z n t
n · Z
1
s −
Zs1+Zs2+...+Zsbn b
n
! =
1 nE
Zt1·Zs1+n−1 n E
Zs1·Zt2− bn nbnE
Zt1·Zs1−bn(n−1) nbn E
Zt2·Zs1=0.
(Of course the index 1 can be replaced byifor any 1≤i≤2m.)
Part (1): First, for anyt>0, the (d·2m-dimensional) vectorYtis independent of the (d-dimensional) vectorZt, because thed(2m+1)-dimensional vector
(Zt,Z1t −Zt,Z2t −Zt, . . . ,Z2 m
t −Zt)T
is normal (since it is a linear transformation of the d · 2m dimensional vector (Zt1,Zt2, . . . ,Zt2m)T, which is normal by Lemma 9), and so it is sufficient to recall that Zt and
Zti−Ztare uncorrelated for 1≤i≤2m.
To complete the proof of (a), recall (2.2) and (2.3) and notice that the conditional distribution of {Zs}s≥t givenFtonly depends on its starting pointZt, as it is that of a Brownian path appropriately slowed down, whateverYt(or, equivalently, whatever Zt=Yt+Zt) is. Since, as we have seen,Yt is independent ofZt, we are done.
Part (2): LetA∈ T. By Dynkin’s Lemma, it is enough to show that(Yt1, ...,Ytk) is independent of
Afor 0 ≤ t1 < ... < tk and k ≥ 1. Since A∈ T ⊂σ(Zs;s≥ tk+1), it is sufficient to show that (Yt
1, ...,Ytk)is independent of{Zs}s≥tk+1.
To see this, similarly as in Part (1), notice that the conditional distribution of{Zs}s≥tk+1givenFtk+1
only depends on its starting pointZtk+1, as it is that of a Brownian path appropriately slowed down,
whatever the vector (Yt
1, ...,Ytk) is. If we show that (Yt1, ...,Ytk) is independent of Ztk+1, we are
done.
To see why this latter one is true, one just have to repeat the argument in (a), using again normality6 and recalling that the vectorsZt andZsi−Zs are uncorrelated.
Remark 15 (Conditioning on the final position of Z). Let N := limt→∞Zt (recall that N ∼
N(0, 2Id)) and
Px(·):=P(· |N= x).
By Lemma 14, Px(Yt ∈ ·) =P(Yt ∈ ·)for almost all x ∈Rd. It then follows that the decomposition
Zt = Zt⊕Yt as well as the result obtained for the distribution ofY in subsections 4.1 and 4.2 are
true underPx too, for almost allx ∈Rd. ⋄
5.2
Main result and a conjecture
So far we have obtained that on the time interval [m,m+1), Y1 corresponds to the Ornstein-Uhlenbeck operator
1 2σ
2
m∆−γx· ∇,
whereσm→1 asm→ ∞, with asymptotically vanishing correlation between the driving Brownian
motions; that
dYt=A(m)dW
(m)
t −γYtdt,
where{Wsm,i, s≥0; i=1, 2, ..., 2m}are independent Brownian motions and r(A(m)) =2m−1, and finally, the independence of the center of mass and the relative motions as in Lemma 14.
6We now need normality for finite dimensional distributions and not just for one dimensional marginals, but this is
We now go beyond these preliminary results and state a theorem (the main result of this article) and a conjecture on the local behavior of the system. Recall that one can considerZnas an element
ofMf(Rd)by putting unit point mass at the site of each particle; with a slight abuse of notation we
will writeZn(dy). Let{Px,x ∈Rd
}be as in Remark 15. Our main result is as follows.
Theorem 16(Scaling limit for the attractive case). Ifγ >0, then, as n→ ∞,
2−nZn(dy) w
⇒
γ
π
d/2
exp−γ|y−x|2dy, Px−a.s. (5.1)
for almost all x∈Rd. Consequently,
2−nE Zn(dy)⇒w fγ(y)dy, (5.2) where
fγ(·) =π(4+γ−1)−d/2exp
−| · |2 4+γ−1
.
Remark 17.
(i) The proof of Theorem 16 will reveal that actually
2−tnZ tn(dy)
w
⇒
γ
π
d/2
exp−γ|y−x|2dy, Px −a.s.
holds for any given sequence{tn}with tn↑ ∞asn→ ∞. This, of course, is still weaker than P-a.s. convergence as t → ∞, but one can probably argue, using the method of Asmussen and Hering, as in Subsection 4.3 of[4]to upgrade it to continuous time convergence. Nev-ertheless, since our model is defined with unit time branching anyway, we felt satisfied with (5.1).
(ii) Notice that fγ, which is the limiting density of the intensity measure, is the density for N
0,
2+21γ
Id
. This is the convolution ofN 0, 2Id, representing the randomness of
the final position of the center of mass (c.f. Lemma 6) and N 0,1 2γ
Id, representing the final distribution of the mass scaled Ornstein-Uhlenbeck branching particle system around its center of mass (c.f. (5.1)). For strong attraction, the contribution of the second term is
negligible. ⋄
Conjecture 18(Dichotomy for the repulsive case). Letγ <0.
1. If|γ| ≥ log 2d , then Z suffers local extinction:
Zn(dy)⇒v 0, P−a.s.
2. If|γ|< log 2d , then
5.3
Discussion of the conjecture
The intuition behind the phase transition at log 2/d is as follows. For a branching diffusion onRd
with motion generator L, smooth nonzero spatially dependent exponential branching rate β(·) ≥ 0 and dyadic branching, it is known (see Theorem 3 in [5]) that either local extinction or local exponential growth takes place according to whetherλc≤0 orλc>0, whereλc=λc(L+β)is the
generalized principle eigenvalue7of L+β onRd. In particular, forβ
≡B>0, the criterion for local exponential growth becomesB>|λc(L)|, whereλc(L)≤0 is the generalized principle eigenvalue of
L, which is also the ‘exponential rate of escape from compacts’ for the diffusion corresponding toL. The interpretation of the criterion in this case is that a large enough mass creation can compensate the fact that individual particles drift away from a given bounded set. (Note that if Lcorrespond to a recurrent diffusion thenλc(L) =0. )
In our case, the situation is similar, withλc=dγfor the outward Ornstein-Uhlenbeck process, taking
into account that for unit time branching, the role of B is played by log 2. The condition for local exponential growth should therefore be log 2>d|γ|.
The scaling 2−ned|γ|n comes from a similar consideration, noting that in our unit time branching setting, 2n replaces the termeβt appearing in the exponential branching case, whileeλc(L)tbecomes
eλc(L)n=edγn.
For a continuous time result analogous to (2) see Example 11 in[4]. Note that since the rescaled (vague) limit of Zn(dy) is translation invariant (i.e. Lebesgue), the final position of the center of mass plays no role.
Although we will not prove Conjecture 18, we will discuss some of the technicalities in section 8.
6
Proof of Theorem 3
Sinceα,βare constant, the branching is independent of the motion, and thereforeN defined by
Nt:=e−βtkXtk
is a nonnegative martingale (positive onS) tending to a limit almost surely. It is straightforward to check that it is uniformly bounded in L2 and is therefore uniformly integrable (UI). Write
Xt=
e−βt〈id,Xt〉 e−βtkX
tk
= e
−βt
〈id,Xt〉 Nt .
We now claim that N∞ > 0 a.s. on S. Let A:= {N∞ = 0}. Clearly S∁ ⊂ A, and so if we show that P(A) = P(S∁), then we are done. As is well known, P(S∁) = e−β/α. On the other hand, a
standard martingale argument (see the argument after formula (20) in[3]) shows that 0≤u(x):= −logPδx(A)must solve the equation
1
2∆u+βu−αu 2=0,
but sincePδx(A) =P(A)constant, therefore−logPδx(A)solvesβu−αu
2=0. SinceNis UI, no mass
is lost in the limit, givingP(A)<1. Sou>0, which in turn implies that−logPδ
Once we know thatN∞>0 a.s. onS , it is enough to focus on the term e−βt〈id,Xt〉: we are going to show that it converges almost surely. Clearly, it is enough to prove this coordinate-wise.
Recall a particular case of theH-transform for the(L,β,α;Rd)-superdiffusionX (see Appendix B in
[7]):
Lemma 19. Define XH by
XHt :=H(·,t)Xt
that is, dX
H t
dXt
=H(·,t)
, t ≥0. (6.1)
If X is an(L,β,α;Rd)-superdiffusion, and H(x,t):=e−λth(x),where h is a positive solution of (L+
β)h=λh, then XH is a(L+a∇hh· ∇, 0,e−λtαh;Rd)-superdiffusion.
In our case β(·) ≡ β. So choosing h(·) ≡ 1 and λ = β, we have H(t) = e−βt and XH is a (1
2∆, 0,e
−βtα;Rd)-superdiffusion, that is, a critical super-Brownian motion with a clock that is
slow-ing down. Since, as noted above, it is enough to prove the convergence coordinate-wise, we can assume thatd=1. One can write
e−βt〈id,Xt〉=〈id,XHt 〉.
Let{Ss}s≥0 be the ‘expectation semigroup’ forX, that is, the semigroup corresponding to the oper-ator 12∆ +β. The expectation semigroup{SsH}s≥0 forXH satisfies Ts :=SsH =e−
βs
Ss and thus it
corresponds to Brownian motion. In particular then
Ts[id] =id. (6.2)
(One can pass from bounded continuous functions to f := id by defining f1 := f1x>0 and f2 := f1x≤0, then noting that by monotone convergence, Eδx〈fi,X
H
t 〉= Exfi(Wt) ∈(−∞,∞), i = 1, 2,
whereW is a Brownian motion with expectationE, and finally taking the sum of the two equations.) ThereforeM:=〈id,XH〉is a martingale:
Eδ
x Mt| Fs
=Eδ
x
〈id,XtH〉 | Fs=
EXs〈id,XHt 〉= Z
R
Eδy〈id,X H t 〉X
H s (dy) =
Z
R
y XsH(dy) =Ms.
We now show that M is UI and even uniformly bounded in L2, verifying its a.s. convergence, and that of the center of mass. To achieve this, definegn by gn(x) =|x| ·1{|x|<n}. Then we have
E〈id,XtH〉2=E|〈id,XtH〉|2≤E〈|id|,XtH〉2,
and by the monotone convergence theorem we can continue with
= lim
n→∞E〈gn,X
H t 〉
2.
Sincegn is compactly supported, there is no problem to use the moment formula and continue with
= lim
n→∞
Z t
0
ds e−βs〈δ0,Ts[αg2n]〉=αnlim→∞
Z t
0
Recall that{Ts;s≥0}is the Brownian semigroup, that is,Ts[f](x) =Exf(Ws), whereW is Brown-ian motion. Sincegn(x)≤ |x|, therefore we can trivially upper estimate the last expression by
α
Z t
0
ds e−βsE0(W2
s ) =α
Z t
0
ds se−βs=α
1−e−βt
β2 − t e−βt
β
< α β2.
Since this upper estimate is independent oft, we are done:
sup
t≥0
E〈id,XtH〉2≤ α
β2.
Finally, we show that X has continuous paths. To this end we first note that we can (and will) consider a version of X where all the paths are continuous in the weak topology of measures. We now need a simple lemma.
Lemma 20. Let{µt, t ≥0}be a family inMf(Rd)and assume that t0>0andµt w
⇒µt0 as t→t0. Assume furthermore that
C=Ct0,ε:=cl
t[0+ε
t=t0−ε
supp(µt)
is compact with someε >0. Let f :Rd
→Rd be a continuous function. Thenlim
t→t0〈f,µt〉=〈f,µt0〉.
Proof of Lemma 20. First, if f = (f1, ...,fd) then all fi are Rd → R continuous functions and
limt→t0〈f,µt〉 = 〈f,µt0〉 simply means that limt→t0〈fi,µt〉 = 〈fi,µt0〉. Therefore, it is enough to prove the lemma for anRd
→R continuous function. Let k be so large that C ⊂ Ik := [−k,k]d. Using a mollified version of1[−k,k], it is trivial to construct a continuous function bf =:Rd→Rsuch that bf = f on Ik and bf =0 onRd
\I2k. Then,
lim
t→t0〈
f,µt〉=tlim
→t0〈 b
f,µt〉=〈fb,µt0〉=〈f,µt0〉,
since bf is a bounded continuous function.
Returning to the proof of the theorem, let us invoke the fact that for
Cs(ω):=cl [
z≤s
supp(Xz(ω)) !
,
we have P(Cs is compact) = 1 for all fixed s ≥ 0 (compact support property; see [6]). By the monotonicity ins, there exists anΩ1⊂ΩwithP(Ω1) =1 such that forω∈Ω1,
Cs(ω)is compact∀s≥0.
Letω∈Ω1and recall that we are working with a continuous path version ofX. Then letting f :=id and µt = Xt(ω), Lemma 20 implies that for t0 > 0, limt→t0〈id,Xt(ω)〉 = 〈id,Xt0(ω)〉. The right
7
Proof of Theorem 16
Before we prove (5.1), we note the following. Fixx ∈Rd. Abbreviate
ν(x)(dy):= γ
π
d/2
exp−γ|y−x|2dy.
Since 2−nZn(dx),ν(x)∈ M1(Rd), therefore, as is well known8, 2−nZ n(dx)
w
⇒ν(x)is in fact equiva-lent to
∀g∈ E : 2−n〈g,Zn〉 → 〈g,ν(x)〉,
whereE is any given family of bounded measurable functions withν(x)-zero (Lebesgue-zero) sets of discontinuity that is separating forM1(Rd).
In fact, one can pick acountableE, which, furthermore, consists of compactly supported functions. Such anE is given by the indicators of sets inR.
Fix such a familyE. SinceE is countable, in order to show (5.1), it is sufficient to prove that for almost allx ∈Rd,
Px(2−n〈g,Zn〉 → 〈g,ν(x)〉) =1, g∈ E. (7.1) We will carry out the proof of (7.1) in several subsections.
7.1
Putting
Y
and
Z
together
The following remark is for the interested reader familiar with [4] only. It can be skipped without any problem.
Remark 21. Once we have the description ofY as in Subsections 4.1 and 4.2 and Remark 15, we can try to put them together with the Strong Law of Large Numbers for the local mass from[4]for the processY.
If the components of Y were independent and the branching rate were exponential, Theorem 6 of[4]would be readily applicable. However, since the 2m components of Y are not independent (as we have seen, their degree of freedom is 2m−1) and since, unlike in [4], we now have unit time branching, the method of [4] must be adapted to our setting. The reader will see that this
adaptation requires quite a bit of extra work. ⋄
Let fe(·) = feγ(·):= (πγ)d/2exp−γ| · |2, and note that ef is the density forN(0,(2γ)−1Id). We now claim that in order to show (7.1), it is enough to prove that for almost allx,
Px(2−n〈g,Yn〉 → 〈g,ef〉) =1, g∈ E. (7.2) This is because
lim
n→∞2 −n
〈g,Zn〉= lim
n→∞2 −n
〈g,Yn+Zn〉= lim
n→∞2 −n
〈g(·+Zn),Yn〉=I+I I,
where
I:= lim
n→∞2 −n
〈g(·+x),Yn〉
and
I I:= lim
n→∞2 −n
〈g(·+Zn)−g(·+x),Yn〉.
Now, (7.2) implies that for almost all x, I =〈g(·+x),ef(·)〉 Px-a.s., while the compact support of g, and Heine’s Theorem yields thatI I=0, Px−a.s. Hence, limn→∞2−n〈g,Zn〉=〈g(·+x),ef(·)〉=
〈g(·),fe(· −x)〉, Px-a.s., giving (7.1).
Next, let us see how (5.1) implies (5.2). Let g be continuous and bounded. Since 2−n〈Zn,g〉 ≤ kgk∞, it follows by bounded convergence that
lim
n→∞E2 −n
〈Zn,g〉=
Z
Rd
Ex
lim
n→∞2 −n
〈Zn,g〉
Q(dx) = Z
Rd
〈g(·),fe(· −x)〉Q(dx),
whereQ∼ N(0, 2Id). Now, if fb∼ N(0, 2Id)then, since fγ ∼ N
0,
2+21γ
Id
, it follows that
fγ= bf ∗ef and Z
Rd
〈g(·),fe(· −x)〉Q(dx) =〈g(·),fγ〉,
yielding (5.2).
Next, notice that it is in fact sufficient to prove (7.2)under P instead of Px. Indeed, by Lemma 14,
Px
lim
n→∞2 −n
〈g,Yn〉=〈g,ef〉
=P
lim
n→∞2 −n
〈g,Yn〉=〈g,ef〉 |N=x
=P
lim
n→∞2 −n
〈g,Yn〉=〈g,fe〉
.
Let us use the shorthandUn(dy):=2−nYt(dy); in generalUt(dy):= 1
ntYt(dy). With this notation,
our goal is to show that
P(〈g,Un〉 → 〈g,fe〉) =1, g∈ E. (7.3)
Now, as mentioned earlier, we may (and will) takeE :=I, whereI is the family of indicators of sets inR. Then, it remains to show that
P
Un(B)→ Z
B
e f(x)dx
=1, B∈ R. (7.4)
7.2
Outline of the further steps
Notation 22. In the sequel{Ft}t≥0will denote the canonical filtration forY, rather than the canon-ical filtration forZ.
The following key lemma (Lemma 23) will play an important role. It will be derived using Lemma 25 and (7.12), where the latter one will be derived with the help of Lemma 25 too. Then, Lemma 23 together with (7.14) will be used to complete the proof of (7.4) and hence, that of Theorem 16.
Lemma 23. Let B⊂Rd be a bounded measurable set. Then,
lim
n→∞
Un+mn(B)−E(Un+mn(B)| Fn)
7.3
Establishing the crucial estimate (7.12) and the key Lemma 23
LetYni denote the “ith” particle at timen,i=1, 2, ..., 2n. SinceBis a fixed set, in the sequel we will simply writeUninstead ofUn(B). Recall the time inhomogeneity of the underlying diffusion process
and note that by the branching property, we have theclumping decomposition: forn,m≥1,
Un+m= 2n X
i=1
2−nUm(i), (7.6)
where givenFn, each member in the collection{Um(i): i=1, ..., 2n}is defined similarly to Um but withYmreplaced by the timemconfiguration of the particles starting atYni, i=1, ..., 2n, respectively, and with motion component 1
2σn+k∆−γx·∆in the time interval[k,k+1).
7.3.1 The functionsa andζ
Next, we define two positive functions,aandζon(1,∞). Here is a rough motivation.
(i) The functiona·will be related (via (7.9) below) to theradial speedof the particle systemY. (ii) The functionζ(·), will be related (via (7.10) below) to thespeed of ergodicityof the underlying
Ornstein-Uhlenbeck process.
Fort>1, define
at :=C0· p
t, (7.7)
ζ(t):=C1logt, (7.8)
whereC0andC1 are positive (non-random) constants to be determined later. Note that
mt:=ζ(at) =C3+C4logt
withC3=C1logC0∈RandC4=C1/2>0. We will use the shorthand
ℓn:=2⌊mn⌋.
Recall that efγ is the density for N(0,(2γ)−1Id) and let q(x,y,t) =q(γ)(x,y,t) andq(x, dy,t) = q(γ)(x, dy,t)denote the transition density and the transition kernel, respectively, corresponding to the operator 12∆−γx· ∇. We are going to show below that for sufficiently large C0 and C1, the following holds. For each givenx ∈Rd andB
⊂Rd nonempty bounded measurable set,
P∃n0,∀n0<n∈N : supp(Yn)⊂Ba
n
=1, and (7.9)
lim
t→∞z∈supBt,y∈B
q(z,y,ζ(t)) e
fγ(y) −1
=0. (7.10)
Remark 9 in [4]), and thus they can be mimicked in our case for the Ornstein-Uhlenbeck process performed by the particles inY (which corresponds to the operator1
2σm∆−γx·∇on[m,m+1), m≥ 1), despite the fact that the particle motions are now correlated. These expectation calculations lead to the estimate that the growth rate of the support ofY satisfies (7.9) with a sufficiently large C0=C0(γ). The same example shows that (7.10) holds with a sufficiently largeC1=C1(γ).
Remark 24. Denote by ν = νγ ∈ M1(Rd) the distribution of
N(0,(2γ)−1Id). Let B ⊂ Rd be a
nonempty bounded measurable set. Takingt=anin (7.10) and recalling thatζ(an) =mn,
lim
n→∞z∈Bsupan,y∈B
q(z,y,mn) e
fγ(y) −1 =0.
Since efγ is bounded, this implies that for any bounded measurable setB⊂Rd,
lim
n→∞zsup∈B
an
q(z,B,mn)−ν(B)=0. (7.11)
We will use (7.11) in Subsection 7.4. ⋄
7.3.2 Covariance estimates
Let{Ymi,nj, j =1, ...,ℓn}be the descendants ofYni at timemn+n. SoY
1,j
mn andY
2,k
mn are respectively
the jth and kth particle at timemn+nof the trees emanating from the first and second particles
at time n. It will be useful to control the covariance between 1B(Ym1,nj) and1B(Y
2,k
mn), where B is a
nonempty, bounded open set. To this end, we will need the following lemma, the proof of which is relegated to Section 9 in order to minimize the interruption in the main flow of the argument.
Lemma 25. Let B⊂Rd be a bounded measurable set.
(a) There exists a non-random constant K(B)such that if C=C(B,γ):= 3
γ|B|
2K(B), then
P
∀n large enough and∀ξ,ξe∈Πn, ξ6=ξe:
P(ξmn,ξemn∈B| Fn)−P(ξmn∈B| Fn)P(ξemn∈B| Fn) ≤ C n
2n
=1,
whereΠndenotes the collection of thoseℓnparticles, which, start at some time-n location of their
parents and run for (an additional) time mn.
(b) Let C=C(B):=ν(B)−(ν(B))2. Then
P
lim
n→∞ξsup∈Π
n Var
1{ξmn∈B}| Fn
−C=0
=1.
Remark 26. In the sequel, instead of writingξmn andξemn, we will use the notationY i1,j
mn andY
i2,k
mn
7.3.3 The crucial estimate (7.12)
LetB⊂Rd be a bounded measurable set andC=C(B,γ)as in Lemma 25. Define
Zi:=
1
ℓn ℓn X
j=1
h
1B(Ymi,j
n)−P(Y i,j
mn∈B|Y i n)
i
, i=1, 2, ..., 2n.
With the help of Lemma 25, we will establish the following crucial estimate, the proof of which is provided in Section 9.
Claim 27. There exists a non-random constantCb(B,γ)>0such that
P
X
1≤i6=j≤2n
EZiZj| Fn
≤Cb(B,γ)nℓn
2n
X
i=1
EZi2| Fn
, for large n’s
=1. (7.12)
The significance of Claim 27 is as follows.
Claim 28. Lemma 25 together with the estimate (7.12) implies Lemma 23.
Proof of Claim 28. Assume that (7.12) holds. By the clumping decomposition under (7.6),
Un+mn−E(Un+mn| Fn) =
2n X
i=1 2−n
Um(i)
n−E(U (i)
mn| Fn)
.
SinceUm(i)
n=ℓ
−1
n
Pℓn j=11B(Y
i,j
mn), therefore
Um(i)
n−E(U (i)
mn| Fn) =U (i)
mn−E(U (i)
mn|Y i n) =Zi.
Hence,
EUn+mn−E(Un+mn| Fn) 2
| Fn
=E 2n X
i=1 2−n
Um(i)
n−E(U (i)
mn| Fn)
2
| Fn
=E 2n X
